What is the value of a Remington Model 10 12-gauge with the serial number on the gun and made in 1903.?

They didn't make the Model 10 until 1908 (-1929), I think your reading a patent date. With a serial number the data can be determined and some of the later production will have a date code stamped on the barrel (anything after 1920). The value is otherwise greatly by condition, condition, and condition if you want a reasonable valuation that must be given.